Determinants and Their Applications in Mathematical Physics

4.3 Skew-Symmetric Determinants 73

=0

λii=(−1)

i+1

[

δi,odd−δi,even

]

=1.

This completes the proofs of the preparatory lemmas. The definition of a

Pfaffian follows. The above lemmas will be applied to prove the theorem

which relates it to a skew-symmetric determinant.

4.3.3 Pfaffians

The nth-order Pfaffian Pfnis defined by the following formula, which

is similar in nature to the formula which defines the determinantAnin

Section 1.2:

Pfn=

∑

sgn

{

1234 ···(2n−1) 2n

i 1 j 1 i 2  2 ··· in jn

}

2 n

ai 1 j 1 ai 2 j 2 ···ainjn,

(4.3.13)

where the sum extends over all possible distinct terms subject to the

restriction

1 ≤is<js≤n, 1 ≤s≤n.. (4.3.14)

Notes on the permutations associated with Pfaffians are given in

Appendix A.2. The number of terms in the sum is

n

∏

s=1

(2s−1) =

(2n)!

2

n

n!

. (4.3.15)

Illustrations

Pf 1 =

∑

sgn

{

12

ij

}

aij (1 term)

=a 12 ,

A 2 = [Pf 1 ]

2

Pf 2 =

∑

sgn

{

1234

i 1 j 1 i 2 j 2

}

ai

1 j 1

ai

2 j 2

(3 terms). (4.3.16)

Omitting the upper parameters,

Pf 2 = sgn{ 1234 }a 12 a 34 + sgn{ 1324 }a 13 a 24 + sgn{ 1423 }a 14 a 23

=a 12 a 34 −a 13 a 24 +a 14 a 23

A 4 = [Pf 2 ]

2

. (4.3.17)

These results agree with (4.3.7) and (4.3.8).